Purely subjective assessment of prior probabilities

In principle, the point is simple, at least in
one-dimensional problem in which there is good perception
of the possible range in which the uncertain variable of interest could
lie: try your best to model your prior beliefs.
In practice, this advice seems difficult to follow because,
even if we have a rough idea of what the value of a quantity
should be, the representation of the prior in mathematical terms
seems very committal,
because a pdf implicitly contains an infinite number
of precise probabilistic statements. (Even the uniform distribution says that we
believe exactly in the same way to all values. Who believes exactly that?)
It is then important to understand that, when expressing priors,
what matters is not the precise mathematical formula,
but the gross value of the probability mass indicated by the formula,
how probabilities are
intuitively perceived and how priors influence posteriors.
When we say, intuitively, we believe something with a 95% confidence,
it means ``we are almost sure,'' but the precise value (95%, instead of 92%
or 98%) is not very relevant. Similarly, when we
say that the prior knowledge is modeled by a Gaussian distribution
centered around with standard deviation
[Eq. (28)], it means
means that we are quite confident that is within
, very sure that it is within
and almost certain that it is
within 3. Values even farther from
are possible, though we do not consider them very likely.
But all models should be taken with a grain of salt,
remembering that they
are often just mathematical conveniences. For example,
a textbook-Gaussian prior includes infinite deviations from the expected value
and even negative values for physical quantities positively defined, like
a temperature or a length. All absurdities, if taken literally.
On the other hand, we think that all experienced physicists
have in mind priors with low probability long tails in order to accommodate
strong deviation from what is expected with highest probability.
(Remember that where the prior is zero, the posterior must also be zero.)

Summing up this point, it is important
to understand that
a prior should tell where the
probability mass is concentrated, without taking too seriously
the details, especially the tails of the distribution
(which should be, however, enough extended to accommodate 'surprises').
The nice feature of Bayes' theorem is the ability of trasform
such vague, fuzzy priors into solid estimates, if a sufficient amount
of good quality data are at hand.
For this reason, the use of improper priors is not considered to be
problematic. Indeed, improper priors
can just be considered a convenient way of modelling relative
beliefs.

In the case we have doubts about the choice of the prior, we can
consider a family of functions with some hyperparameters.
If we worry about the effect of the chosen prior
on the posterior, we can perform a
sensitivity analysis, i.e. to repeat the
analysis for different, reasonable choices of the
prior and check the variation of the result.
The final uncertainty could, then, take into account
also the uncertainty on the prior. Finally, in extreme cases in which
priors play a crucial role and could dramatically change the conclusions,
one should refrain to give probabilistic result, providing, instead, only
Bayes factors, or even just likelihoods. An example of a recent
result about gravitational wave searches presented in this way, see
Astone et al (2002).

Having clarified meaning and role of priors, it is rather evident
that the practical choice of a prior depends on what is appropriate
for the application.
For example, in the area of imaging,
smoothness of a reconstructed image might be appropriate in
some situations. Smoothness may be imposed by a variety
of means, for example, by simply setting the logarithm of
the prior equal to an integral of the square of the second derivative
of the image (von der Linden et al 1996b).
A more sophisticated approach goes under the name of Markov random
fields (MRF), which can even preserve sharp edges in the estimated images
(Bouman and Sauer 1993, Saquib et al 1997). A similar kind of prior
is often appropriate for defomable geometric models, which can be used
to represent the boundaries between various regions, for example,
organs in medical images (Cunningham et al 1998).

A procedure that helps in choosing the prior, expecially important
in the cases in which the
parameters do not have a straightforwardly perceptible
influence on data, is to build a prior predictive pdf and
check if this pdf would produce data conform with our prior beliefs.
The prior predictive distribution is the analogue of the
(posterior) predictive distribution we met in
Sect. 5.7, with
replaced by
(note that the example of
Sect. 5.7 was one-dimensional, with and ),
i.e.
.

Often, expecially in complicated data analyses, we are not sufficiently
knowledgable about the details of the problem. Thus, informative priors
have to be modelled that capture the judgement of experts. For
example, Meyer and Booker (2001) show a formal process of
prior elicitation which has the aim at reducing,
as much as possible, the bias in the experts' estimates
of their confidence limits.
This approach allows one to combine the results from several experts.
In short, we can suggest the use of the
`coherent bet' (Sect. 2)
to force experts to access
their values of probability, asking them to provide an interval in which
they feel `practically sure',
intervals on which they could wager 1:1, and so on.